Numerical linear algebra and random matrix theory have long been coupled, going (at least) back to the seminal work of Goldstine and von Neumann (1951) on the condition number of random matrices. The connections have since gone deeper. A number of authors have noted that matrix factorization algorithms can be applied to Gaussian random matrices with an exact (distributional) characterization of the output. This includes the LU, QR and Schur factorizations and the Golub-Kahan bidiagonalization procedure. In recent years, more attention has been paid to how iterative algorithms act on random matrices and, in particular, whether or not the behavior is universal, i.e., Does the behavior persist for non-Gaussian random matrices? In this talk, I will discuss some of the matrix factorization history and discuss some recent (universal) results on the ``typical" behavior of iterative algorithms from numerical linear algebra. This is joint work with Percy Deift (NYU), Govind Menon (Brown) and Elliot Paquette (OSU).